INteraction between probability and analysis on disordered media

1.构造了无序介质中的扩散过程，并研究了无序介质个数可数时该过程的性质，得到了新的结果。当每种介质上的Dirichlet形式的区域都是Besov空间时，我们应用Besov空间的理论，构造了穿透扩散过程。我们还得到了该过程的热核的一个短时渐近行为，并利用能量函数的变分公式的解来表示“最可几路径”。这是与B.M.Hamble的合作作品，并出现在Probab上。理论上来说。2.构造了d-集(分形集)上的跳跃型过程，得到了该过程的详细热核估计。Dirichlet形式的相应区域又是Besov空间，Besov空间的迹公式是非常有用的。使用概率技术来获得非对角估计。利用热核估计，我们证明了过程的瞬时性/常返性，并计算了过程值域的Hausdorff维。这是与Z.Q.Chen的合作作品，出现在Stoch上。程序研究了自相似集上扩散过程的热核的渐近行为。我们证明了该测度的体积倍增性质与一些非对角热核估计之间的等价性。此外，我们还证明了逃逸时间的某些估计加上局部Nash不等式与上面的某种热核估计是等价的。

1.We have obtained new results for the problem to construct diffusion processes penetrating disordered media and to study properties of the processes when there are countable number of disordered media on a space. When domains of Dirichlet forms on each medium are Besov spaces, we apply the theory of Besov spaces and construct the penetrating diffusion process. We also obtain a short time asymptotic behavior of the heat kernel for the process and express the "most probable path" by means of the solution of variational formula of energy functions. This is a joint work with B.M.Hambly and appears in Probab. Theory Relat. Fields.2.We have constructed jump-type processes on d-sets (fractal sets) and obtained detailed heat kernel estimates of the processes. The corresponding domains of Dirichlet forms are again Besov spaces and the trace formula of Besov spaces are very useful. Probabilistic techniques are used to obtain the off-diagonal estimates. Using the heat kernel estimates, we show transience/recurrence of the process and computed the Hausdorff dimension of the range of the process. This is a joint work with Z.Q.Chen and appears in Stoch. Proc. Their Appl.3.We have studied the asymptotic behavior of the heat kernels for diffusion processes on self-similar sets. We prove an equivalence between the volume doubling property of the measure and some on-diagonal heat kernel estimates. Furthermore, we prove an equivalence between some estimates of escaping times plus local Nash inequalities and some type of heat kernel estimates from above.